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<li><p>Different types of boundary conditions would give very different eigenvalues and eigenfunctions.</p></li>
<li><p>In these examples, the eigenfunctions are sine and cosine functions, in the same form as the trig set we use in Fourier series. Recall that the trig set is a mutually orthogonal set. So, for each of these eigenvalue problems, the set of eigenfunctions are mutually orthogonal. In fact, this is a more general property for eigenfunctions. One can define proper inner product such that eigenfunctions for the same eigenvalue problem would always form a mutually orthogonal set.</p></li>
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